The cavity approach for Steiner trees packing problems

The Belief Propagation approximation, or cavity method, has been recently applied to several combinatorial optimization problems in its zero-temperature implementation, the max-sum algorithm. In particular, recent developments to solve the edge-disjoint paths problem and the prize-collecting Steiner tree problem on graphs have shown remarkable results for several classes of graphs and for benchmark instances. Here we propose a generalization of these techniques for two variants of the Steiner trees packing problem where multiple "interacting" trees have to be sought within a given graph. Depending on the interaction among trees we distinguish the vertex-disjoint Steiner trees problem, where trees cannot share nodes, from the edge-disjoint Steiner trees problem, where edges cannot be shared by trees but nodes can be members of multiple trees. Several practical problems of huge interest in network design can be mapped into these two variants, for instance, the physical design of Very Large Scale Integration (VLSI) chips. The formalism described here relies on two components edge-variables that allows us to formulate a massage-passing algorithm for the V-DStP and two algorithms for the E-DStP differing in the scaling of the computational time with respect to some relevant parameters. We will show that one of the two formalisms used for the edge-disjoint variant allow us to map the max-sum update equations into a weighted maximum matching problem over proper bipartite graphs. We developed a heuristic procedure based on the max-sum equations that shows excellent performance in synthetic networks (in particular outperforming standard multi-step greedy procedures by large margins) and on large benchmark instances of VLSI for which the optimal solution is known, on which the algorithm found the optimum in two cases and the gap to optimality was never larger than 4 %

Bicriteria Network Design Problems

We study a general class of bicriteria network design problems. A generic problem in this class is as follows: Given an undirected graph and two minimization objectives (under different cost functions), with a budget specified on the first, find a <subgraph \from a given subgraph-class that minimizes the second objective subject to the budget on the first. We consider three different criteria - the total edge cost, the diameter and the maximum degree of the network. Here, we present the first polynomial-time approximation algorithms for a large class of bicriteria network design problems for the above mentioned criteria. The following general types of results are presented. First, we develop a framework for bicriteria problems and their approximations. Second, when the two criteria are the same %(note that the cost functions continue to be different) we present a ``black box'' parametric search technique. This black box takes in as input an (approximation) algorithm for the unicriterion situation and generates an approximation algorithm for the bicriteria case with only a constant factor loss in the performance guarantee. Third, when the two criteria are the diameter and the total edge costs we use a cluster-based approach to devise a approximation algorithms --- the solutions output violate both the criteria by a logarithmic factor. Finally, for the class of treewidth-bounded graphs, we provide pseudopolynomial-time algorithms for a number of bicriteria problems using dynamic programming. We show how these pseudopolynomial-time algorithms can be converted to fully polynomial-time approximation schemes using a scaling technique.Comment: 24 pages 1 figur

On Embeddability of Buses in Point Sets

Set membership of points in the plane can be visualized by connecting corresponding points via graphical features, like paths, trees, polygons, ellipses. In this paper we study the \emph{bus embeddability problem} (BEP): given a set of colored points we ask whether there exists a planar realization with one horizontal straight-line segment per color, called bus, such that all points with the same color are connected with vertical line segments to their bus. We present an ILP and an FPT algorithm for the general problem. For restricted versions of this problem, such as when the relative order of buses is predefined, or when a bus must be placed above all its points, we provide efficient algorithms. We show that another restricted version of the problem can be solved using 2-stack pushall sorting. On the negative side we prove the NP-completeness of a special case of BEP.Comment: 19 pages, 9 figures, conference version at GD 201

A polylogarithmic approximation algorithm for group Steiner tree problem

The group Steiner tree problem is a generalization of the Steiner tree problem where we are given several subsets (groups) of vertices in a weighted graph, and the goal is to find a minimum-weight connected subgraph containing at least one vertex from each group. The problem was introduced by Reich and Widmayer and finds applications in VLSI design. The group Steiner tree problem generalizes the set covering problem, and is therefore at least as hard. We give a randomized $O(\log^3 n \log k)$-approximation algorithm for the group Steiner tree problem on an $n$-node graph, where $k$ is the number of groups.The best previous performance guarantee was $(1+\frac{\ln k}{2})\sqrt{k}$ (Bateman, Helvig, Robins and Zelikovsky). Noting that the group Steiner problem also models the network design problems with location-theoretic constraints studied by Marathe, Ravi and Sundaram, our results also improve their bicriteria approximation results. Similarly, we improve previous results by Slav{\'\i}k on a tour version, called the errand scheduling problem. We use the result of Bartal on probabilistic approximation of finite metric spaces by tree metrics to reduce the problem to one in a tree metric. To find a solution on a tree, we use a generalization of randomized rounding. Our approximation guarantees improve to $O(\log^2 n \log k)$ in the case of graphs that exclude small minors by using a better alternative to Bartal's result on probabilistic approximations of metrics induced by such graphs (Konjevod, Ravi and Salman) -- this improvement is valid for the group Steiner problem on planar graphs as well as on a set of points in the 2D-Euclidean case

An Improved Augmented Line Segment based Algorithm for the Generation of Rectilinear Steiner Minimum Tree

An improved Augmented Line Segment Based (ALSB) algorithm for the construction of Rectilinear Steiner Minimum Tree using augmented line segments is proposed. The proposed algorithm works by incrementally increasing the length of line segments drawn from all the points in four directions. The edges are incrementally added to the tree when two line segments intersect. The reduction in cost is obtained by postponing the addition of the edge into the tree when both the edges (upper and lower L-shaped layouts) are of same length or there is no overlap. The improvement is focused on reduction of the cost of the tree and the number of times the line segments are augmented. Instead of increasing the length of line segments by 1, the line segments length are doubled each time until they cross the intersection point between them. The proposed algorithm reduces the wire length and produces good reduction in the number of times the line segments are incremented. Rectilinear Steiner Minimum Tree has the main application in the global routing phase of VLSI design. The proposed improved ALSB algorithm efficiently constructs RSMT for the set of circuits in IBM benchmark

An asymptotically optimal push-pull method for multicasting over a random network

We consider allcast and multicast flow problems where either all of the nodes or only a subset of the nodes may be in session. Traffic from each node in the session has to be sent to every other node in the session. If the session does not consist of all the nodes, the remaining nodes act as relays. The nodes are connected by undirected links whose capacities are independent and identically distributed random variables. We study the asymptotics of the capacity region (with network coding) in the limit of a large number of nodes, and show that the normalized sum rate converges to a constant almost surely. We then provide a decentralized push-pull algorithm that asymptotically achieves this normalized sum rate without network coding.Comment: 13 pages, extended version of paper presented at the IEEE International Symposium on Information Theory (ISIT) 2012, minor revision to text to address review comments, to appear in IEEE Transactions in information theor